Abstract
In “Is Logic Empirical” (Putnam 1968), Putnam formulates an empirical argument against classical logic—in particular, an apparent counterexample to the distributivity laws. He argues further that this argument is also an argument in favor of quantum logic. Here we challenge this second conclusion, arguing instead that counterexamples in logic are counterexamples not to particular inferences, but to logics as a whole. The key insight underlying this argument is that what counts as a legitimate translation from natural language to formal language is dependent on the background logic being assumed. Hence, in the face of a counterexample, one can move to a logic that fails to validate the inference seemingly counter-instanced, or one can move to a logic where the best translation of the natural language claims involved in the counterexample are no longer best translated as an instance of the inference in question.
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Notes
- 1.
Although arguments for the in-principle possibility of purely empirical challenges to the correctness of a logic – classical or otherwise – go back to Quite (1951), Putnam was amongst the first to provide a serious putative example of such an empirical challenge.
- 2.
Of course, I personally don’t believe that there is one correct logic (i.e. I am a logical pluralist of some sort), nor do even I believe that quantum logic is a plausible candidate for being one of the multitude of ‘correct’ or ‘best’ logics – see Cook (2014). The point is that nothing in the present paper depends on these further views.
- 3.
There are, of course, a number of extant criticisms of his argument along these lines. There are four primary themes running through such criticisms. First, there are objections to the role that realism plays in Putnam’s argument – for a prominent example of this sort, see Dummett (1976). Second, there are objections to Putnam’s presentation of the physics, the logic, or the connections between the two – for a recent example of this sort, see Maudlin (2005). Third, there are objections to the claim that formal quantum logic – that is, the propositional/first-order theory obtained via constructing a semantics in terms of the lattice of ‘quantum propositions’ – blocks the problematic inferences anyway. Gardner (1971) and Gibbons (1987) are notable examples of this approach. Finally, there is the claim – forcefully argued for in Hellman (1980) – that the language and logic within which quantum mechanics is formulated is fully classical, and hence shifting to a different logic once these puzzles arise (regardless of whether, pace Gardner (1971) and Gibbons (1987), such a logic actually blocks the problematic inferences) amounts to ignoring the problem rather than addressing it. The present essay will address none of these specific concerns.
- 4.
Of course, a fully sufficient examination of whether or not quantum mechanics really does provide counterexamples to classical logic would require a significantly deeper understanding of the relevant science than I possess. Fortunately, as I have already emphasized, the point of this paper, which really concerns the methodology of logic rather than that of science, does not depend on answering this distinct question.
- 5.
Of course, it goes without saying that any novel conclusions we arrive at with respect to the correct methodology for dealing with purported counterexamples to our favored logics will have very real consequences for what candidate logics we might take seriously as correct or (if one has pluralist leanings) legitimate. The point is merely that we are not focusing here on answering the latter question.
- 6.
The fact that we are granting all of this for the sake of argument also, it is hoped, goes some ways towards excusing the looseness of the science in this section!
- 7.
There are two reasons for doing so: First, as already noted, the point is not to show that we cannot arrive at quantum logic via a correct application of \(\mathsf{CARL}\), but merely that quantum logic is not the only destination that we might arrive at via application of that pattern of argumentation. Second, it seems quite plausible to me that, from within the context of quantum logic, these are (up to logical equivalence within \(\mathsf{Q}\)) the best translations of the claims in question!
- 8.
And perhaps for good pedagogical reasons, irrespective of what one’s final view is on the correct logic or logics!
- 9.
That is, “\(\Phi \) unless \(\Psi \)” ought to be logically equivalent to “\(\Psi \) unless \(\Phi \)”. This is not implied by the validity of the two rules mentioned above, but seems like a plausible additional constraint on formalizations of “unless” – one that undergraduate logic students would likely agree to relatively easily.
- 10.
I am not suggesting that there might not be other criteria that would allow us to decide between these two distinct translations of the natural language expression “unless” in intuitionistic contexts. But I do not know what such considerations might look like, other than, perhaps, an empirical analysis of the actual inferential practices of intuitionistists with respect to “unless”.
- 11.
This is not to say that there are no reasons to prefer one formalization over another, logically equivalent one. After all, one might be syntactically much simpler than the other. The point is that, insofar as the primary criterion for successful formalization is getting the truth conditions right – at least, in situations like the present one where the issue is whether one claim follows logically from another – we have no reason to prefer one formalization over another logically equivalent one.
- 12.
For more on how we might read the intuitionistic connectives as synonymous with the corresponding classical connectives while nevertheless tracking determinacy or knowability in a manner in which their classical counterparts do not, see Cook (2014).
- 13.
Note that we need not restrict our attention to this handful of simple translations. There are many other interesting, disjunction-like operators definable within intuitionistic logic. Interesting examples include pseudo-disjunction :
$$\begin{aligned} B_1 \dot{\vee } B_2 =_{df} ((B_1 \rightarrow B_2) \rightarrow B_2) \wedge ((B_2 \rightarrow B_1) \rightarrow B_1) \end{aligned}$$Church disjunction :
$$\begin{aligned} B_1 \dddot{\vee } B_2 =_{df} (B_1 \rightarrow B_2) \rightarrow ((B_2 \rightarrow B_1) \rightarrow B_1) \end{aligned}$$and Cornish disjunction :
$$\begin{aligned} B_1 \star B_2 =_{df} (((B_1 \rightarrow B_2) \rightarrow B_2) \rightarrow B_1) \rightarrow B_1 \end{aligned}$$These are examined in detail in Humberstone (2011) – the absolutely definitive and authoritative study of propositional connectives in classical and non-classical logics – on pages 555, 235, and 235 respectively.
- 14.
This last bit is formulated a bit obscurely so as not to prejudice which formalization we, in the end, prefer.
- 15.
Of course, a full defense of an intuitionistic approach to the puzzle would also need to provide an intuitionistic account of probability theory – one that, in particular, allowed for classically equivalent claims to receive distinct probabilities. After all, a full and careful presentation of the puzzle in question – such as the one given by Putnam himself in Putnam (1968) – presents the issue in terms of probabilities and not in terms of truth and falsity. Since the purpose here is not to provide a full-fledged defense of the intuitionistic approach to this puzzle, but merely to demonstrate the existence and potential viability of methodological alternatives such as, but not limited to, the intuitionistic approach, I shall not pursue such an account here. For a recent account of intuitionistic probability that might do the job, however, the reader is encouraged to consult (Weatherson 2003).
- 16.
It is assumed that the logics \(\mathcal {L}_1\) and \(\mathcal {L}_2\) are built on the same language.
- 17.
For simplicity of discussion I set aside possible logical pluralists who accept a multitude of paraconsistent or dialethic logics, and as a result might accept the problematic inference in question.
- 18.
As a former teacher succinctly put the point when I expressed sympathy to something like the intuitonistic version of pluralism sketched here and developed in more detail in Cook (2014):
So let me get this straight. Now you’re a logical pluralist, but classical logic doesn’t even get to be one of the correct logics?
- 19.
In fact, it is consistent in Gödel-Dummett logic, which results from supplementing intuitionistic logic with the linearity axiom:
$$\begin{aligned} (\Phi \rightarrow \Psi ) \vee (\Psi \rightarrow \Phi ) \end{aligned}$$ - 20.
It should be emphasized that there is nothing in \(\mathsf{CARL}\) that privileges or prefers intuitionistic logic as opposed to other non-classical logics. But I am most familiar with, and most sympathetic to, intuitionistic logic and similarly constructive intermediate logics between \(\mathsf{H}\) and \(\mathsf{C}\). Hence I leave the construction of additional examples that use non-classical logics other than intuitionistic logic to the ambitious reader.
In addition, there is nothing in \(\mathsf{CARL}\) that requires that there be a single logic and translation manual that is correct or best. Instead, in the face of an apparent counterexample to classical logic such as the puzzles examined here, there might be multiple, equally good ways of changing both one’s logic and one’s translations of the problematic statements.
- 21.
If the reader would like another possible example, consider a situation where we have a notion of some series of objects being “constructed”, which we can formalize as \(\mathcal {C}(x)\) and understand along the lines of familiar accounts in the philosophy of mathematics that involve notions such as potential infinity or constructivity. In such cases we might want to say that that:
$$\begin{aligned} \text {It is not the case (at any particular time) that every object is constructed}. \end{aligned}$$which the classical logician might (quite correctly, from her perspective) formalize as:
$$\begin{aligned} \lnot (\forall x)(\mathcal {C}(x)) \end{aligned}$$yet we might also want to claim that:
$$\begin{aligned} \text {No object fails (forever) to be constructed}. \end{aligned}$$which the classical logician might (again, correctly) formalize as:
$$\begin{aligned} (\forall x)(\mathcal {C}(x)) \end{aligned}$$Of course, these claims are inconsistent in classical logic, but acceptance of this pair of natural language claims this need not imply a rejection of:
$$\begin{aligned} \lnot (\forall x)(\mathcal {C}(x)), (\forall x)(\mathcal {C}(x)) \vdash _\mathcal {L} \bot \end{aligned}$$(which is, after all, just a complex instance of the standard negation elimination rule). Instead, we might move to intuitionistic logic, but argue that the best translation of the second natural language claim is not the one given above, but is instead:
$$\begin{aligned} (\forall x)(\lnot \lnot \mathcal {C}(x)) \end{aligned}$$Of course, the legitimacy of this move depends on the fact that:
$$\begin{aligned} \lnot (\forall x)(\mathcal {C}(x)), (\forall x)(\lnot \lnot \mathcal {C}(x)) \nvdash \,\,{}_\mathsf{H}\;\bot \end{aligned}$$ - 22.
The reader interested in seeing another attempt to formulate and defend an intuitionistic response to the puzzles of quantum mechanics – one that combines the insights and machinery of intuitionistic logic and quantum logic – is encouraged to consult (Caspers et al. 2009).
- 23.
If the reader is interested in learning more about what, exactly, the necessity of logical consequence amounts to, and how Tarski himself understood this notion, the reader will find no better place to begin than Etchemendy (1999).
- 24.
\(\Phi [\Psi /\Gamma ]\) is the formula that results from uniformly replacing every occurrence of the (primitive) expression \(\Psi \) in \(\Phi \) with \(\Gamma \). \(\Delta [\Psi /\Gamma ]\) is:
$$\begin{aligned} \{ \Theta [\Phi /\Gamma ] : \Theta \in \Delta \} \end{aligned}$$ - 25.
It is no accident that algebraic semantics for logics usually involve either exactly this construction, or something closely akin to it. See, e.g., Dunn and Hardegree (2001) for a standard treatment of algebraic semantics for logics. Also note that the account developed here – and in particular this construction of equivalence classes of formulas – depends on the logic in question satisfying the standard structural rules. Similar, albeit much more mathematically sophisticated, constructions that will support similar arguments for substructural logics are possible, but are left to the reader interested in such things.
- 26.
Of course, we could also replace the sentences of the natural language with equivalence classes of logically equivalent natural language sentences. The methodological problem with this approach, of course, is that our primary means by which to determine which pairs of natural language sentences are logically equivalent is to first determine which logic we will take to be the correct logic, and then project logical equivalence from the formal logic to natural language via the inverse of our translation function. This is the reason we study formal languages in the first place: they are more amenable to mathematical study and manipulation than their natural language counterparts, and thus it is usually easier to delineate philosophically relevant phenomena like the logical consequence relation in the formal sphere, and then project them to natural language, than it is to detect those same phenomena directly in natural language. That being said, if there were some means for identifying, in general, which pairs statements in natural language were logically equivalent to each other that was independent of the methods and tools of formal logic, then we could replace natural language sentences with their equivalence classes in the treatment above. Nothing significant would change with respect to the issues being examined here.
- 27.
Note that the fact that the classical logician and the intuitionistic logician are faced with different possibilities with respect to translating “unless” does not automatically entail that they mean different things by “unless”. Translation, in the context of constructing a formal logic that correctly codifies the natural language consequence relation, need only preserve logical form. But although synonymous statements presumably have the same logical form, the converse is not obviously true. Thus, the classical logician could treat both options are equally legitimate given that her purpose is to study logical consequence, but agree with the intuitionistic logician that these formulas mean different things. See Cook (2014) for more discussion.
- 28.
It is worth noting that standard pedagogical practice in introduction to logic classes violates this methodological observation: typically, in such courses, we introduce the formal language, then teach students how to translate between natural language and our formal language, and only then, after mastering the translation rules we have inculcated in them, do we ask them to consider which rules are and are not valid.
- 29.
Thanks are owed to Geoffrey Hellman, Stewart Shapiro, and Jos Uffink for helpful conversations on matters related to this paper.
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Cook, R.T. (2018). Logic, Counterexamples, and Translation. In: Hellman, G., Cook, R. (eds) Hilary Putnam on Logic and Mathematics. Outstanding Contributions to Logic, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-96274-0_3
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